Abstract

The effects of birefringence on the light distribution in the focal region of a high-NA optical system are investigated with use of the Debye approach to vector diffraction theory. The attention is limited to uniaxially birefringent media with symmetry axis along the optical axis of the imaging system. The radially (p) and tangentially (s) polarized fields in the exit pupil produce spots in the focal region that are defocused with respect to each other. For small birefringence values the relative defocus causes a distortion and broadening of the spot; for larger values the two spots separate completely. As a corollary to the theory it is shown that there is a tangential tornadolike flow of energy in the focal region when the polarization in the entrance pupil is elliptical.

© 2001 Optical Society of America

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  1. T. Narahara, S. Kobayashi, M. Hattori, Y. Shimpuku, G. J. van den Enden, J. A. H. M. Kahlman, M. van Dijk, R. van Woudenberg, “Optical disc system for digital video recording,” Jpn. J. Appl. Phys. 39, 912–919 (2000).
    [CrossRef]
  2. E. Wolf, “Electromagnetic diffraction in optical systems I. an integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
    [CrossRef]
  3. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [CrossRef]
  4. A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
    [CrossRef]
  5. A. Boivin, J. Dow, E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,” J. Opt. Soc. Am. 57, 1171–1175 (1967).
    [CrossRef]
  6. M. Mansuripur, “Distribution of light at and near the focus of high-numerical-aperture objectives,” J. Opt. Soc. Am. A 3, 2086–2093 (1986).
    [CrossRef]
  7. M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Am. A 6, 786–805 (1989).
    [CrossRef]
  8. H. Ling, S.-W. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A 1, 965–973 (1984).
    [CrossRef]
  9. P. Török, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
    [CrossRef]
  10. P. Török, P. Varga, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. I,” J. Opt. Soc. Am. A 12, 2136–2144 (1995).
    [CrossRef]
  11. P. Török, P. Varga, G. Németh, “Analytical solution of the diffraction integrals and interpretation of wave-front distortion when light is focused through a planar interface between materials of mismatched refractive indices,” J. Opt. Soc. Am. A 12, 2660–2671 (1995).
    [CrossRef]
  12. S. H. Wiersma, T. D. Visser, “Defocusing of a convergingelectromagnetic wave by a plane dielectric interface,” J. Opt. Soc. Am. A 13, 320–325 (1996).
    [CrossRef]
  13. S. H. Wiersma, P. Török, T. D. Visser, P. Varga, “Comparison of different theories for focusing through a plane interface,” J. Opt. Soc. Am. A 14, 1482–1490 (1997).
    [CrossRef]
  14. V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab: I. exact and asymptotic results,” Pure Appl. Opt. 6, 33–52 (1997).
  15. D. G. Flagello, T. Milster, A. E. Rosenbluth, “Theory of high-NA imaging in homogeneous thin films,” J. Opt. Soc. Am. A 13, 53–64 (1996).
    [CrossRef]
  16. J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
    [CrossRef]
  17. D. Jiang, J. J. Stamnes, “Numerical and asymptotic results for focusing of two-dimensional waves in uniaxial crystals,” Opt. Commun. 163, 55–71 (1999).
    [CrossRef]
  18. D. Jiang, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves into uniaxial crystals,” Opt. Commun. 174, 321–334 (2000).
    [CrossRef]
  19. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  20. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK1986).
  21. M. Mansuripur, “Distribution of light at and near the focus of high-numerical-aperture objectives: erratum; Certain computational aspects of vector diffraction problems: erratum,” J. Opt. Soc. Am. A 10, 382–383 (1993).
    [CrossRef]
  22. A. B. Marchant, “Cover sheet aberrations in optical recording,” in Optical Disk Systems and Applications, E. V. LaBudde, ed., Proc. SPIE421, 43–49 (1983).
    [CrossRef]
  23. M. Mansuripur, The Physical Principles of Magneto-Optical Recording (Cambridge U. Press, Cambridge, UK, 1995).

2000 (2)

T. Narahara, S. Kobayashi, M. Hattori, Y. Shimpuku, G. J. van den Enden, J. A. H. M. Kahlman, M. van Dijk, R. van Woudenberg, “Optical disc system for digital video recording,” Jpn. J. Appl. Phys. 39, 912–919 (2000).
[CrossRef]

D. Jiang, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves into uniaxial crystals,” Opt. Commun. 174, 321–334 (2000).
[CrossRef]

1999 (1)

D. Jiang, J. J. Stamnes, “Numerical and asymptotic results for focusing of two-dimensional waves in uniaxial crystals,” Opt. Commun. 163, 55–71 (1999).
[CrossRef]

1998 (1)

J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
[CrossRef]

1997 (2)

S. H. Wiersma, P. Török, T. D. Visser, P. Varga, “Comparison of different theories for focusing through a plane interface,” J. Opt. Soc. Am. A 14, 1482–1490 (1997).
[CrossRef]

V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab: I. exact and asymptotic results,” Pure Appl. Opt. 6, 33–52 (1997).

1996 (2)

1995 (3)

1993 (1)

1989 (1)

1986 (1)

1984 (1)

1967 (1)

1965 (1)

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

1959 (2)

E. Wolf, “Electromagnetic diffraction in optical systems I. an integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Boivin, A.

A. Boivin, J. Dow, E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,” J. Opt. Soc. Am. 57, 1171–1175 (1967).
[CrossRef]

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

Booker, G. R.

Dhayalan, V.

V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab: I. exact and asymptotic results,” Pure Appl. Opt. 6, 33–52 (1997).

Dow, J.

Flagello, D. G.

Hattori, M.

T. Narahara, S. Kobayashi, M. Hattori, Y. Shimpuku, G. J. van den Enden, J. A. H. M. Kahlman, M. van Dijk, R. van Woudenberg, “Optical disc system for digital video recording,” Jpn. J. Appl. Phys. 39, 912–919 (2000).
[CrossRef]

Jiang, D.

D. Jiang, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves into uniaxial crystals,” Opt. Commun. 174, 321–334 (2000).
[CrossRef]

D. Jiang, J. J. Stamnes, “Numerical and asymptotic results for focusing of two-dimensional waves in uniaxial crystals,” Opt. Commun. 163, 55–71 (1999).
[CrossRef]

J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
[CrossRef]

Kahlman, J. A. H. M.

T. Narahara, S. Kobayashi, M. Hattori, Y. Shimpuku, G. J. van den Enden, J. A. H. M. Kahlman, M. van Dijk, R. van Woudenberg, “Optical disc system for digital video recording,” Jpn. J. Appl. Phys. 39, 912–919 (2000).
[CrossRef]

Kobayashi, S.

T. Narahara, S. Kobayashi, M. Hattori, Y. Shimpuku, G. J. van den Enden, J. A. H. M. Kahlman, M. van Dijk, R. van Woudenberg, “Optical disc system for digital video recording,” Jpn. J. Appl. Phys. 39, 912–919 (2000).
[CrossRef]

Laczik, Z.

Lee, S.-W.

Ling, H.

Mansuripur, M.

Marchant, A. B.

A. B. Marchant, “Cover sheet aberrations in optical recording,” in Optical Disk Systems and Applications, E. V. LaBudde, ed., Proc. SPIE421, 43–49 (1983).
[CrossRef]

Milster, T.

Narahara, T.

T. Narahara, S. Kobayashi, M. Hattori, Y. Shimpuku, G. J. van den Enden, J. A. H. M. Kahlman, M. van Dijk, R. van Woudenberg, “Optical disc system for digital video recording,” Jpn. J. Appl. Phys. 39, 912–919 (2000).
[CrossRef]

Németh, G.

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Rosenbluth, A. E.

Shimpuku, Y.

T. Narahara, S. Kobayashi, M. Hattori, Y. Shimpuku, G. J. van den Enden, J. A. H. M. Kahlman, M. van Dijk, R. van Woudenberg, “Optical disc system for digital video recording,” Jpn. J. Appl. Phys. 39, 912–919 (2000).
[CrossRef]

Stamnes, J. J.

D. Jiang, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves into uniaxial crystals,” Opt. Commun. 174, 321–334 (2000).
[CrossRef]

D. Jiang, J. J. Stamnes, “Numerical and asymptotic results for focusing of two-dimensional waves in uniaxial crystals,” Opt. Commun. 163, 55–71 (1999).
[CrossRef]

J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
[CrossRef]

V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab: I. exact and asymptotic results,” Pure Appl. Opt. 6, 33–52 (1997).

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK1986).

Török, P.

van den Enden, G. J.

T. Narahara, S. Kobayashi, M. Hattori, Y. Shimpuku, G. J. van den Enden, J. A. H. M. Kahlman, M. van Dijk, R. van Woudenberg, “Optical disc system for digital video recording,” Jpn. J. Appl. Phys. 39, 912–919 (2000).
[CrossRef]

van Dijk, M.

T. Narahara, S. Kobayashi, M. Hattori, Y. Shimpuku, G. J. van den Enden, J. A. H. M. Kahlman, M. van Dijk, R. van Woudenberg, “Optical disc system for digital video recording,” Jpn. J. Appl. Phys. 39, 912–919 (2000).
[CrossRef]

van Woudenberg, R.

T. Narahara, S. Kobayashi, M. Hattori, Y. Shimpuku, G. J. van den Enden, J. A. H. M. Kahlman, M. van Dijk, R. van Woudenberg, “Optical disc system for digital video recording,” Jpn. J. Appl. Phys. 39, 912–919 (2000).
[CrossRef]

Varga, P.

Visser, T. D.

Wiersma, S. H.

Wolf, E.

A. Boivin, J. Dow, E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,” J. Opt. Soc. Am. 57, 1171–1175 (1967).
[CrossRef]

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

E. Wolf, “Electromagnetic diffraction in optical systems I. an integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (10)

M. Mansuripur, “Distribution of light at and near the focus of high-numerical-aperture objectives,” J. Opt. Soc. Am. A 3, 2086–2093 (1986).
[CrossRef]

M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Am. A 6, 786–805 (1989).
[CrossRef]

H. Ling, S.-W. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A 1, 965–973 (1984).
[CrossRef]

P. Török, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
[CrossRef]

P. Török, P. Varga, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. I,” J. Opt. Soc. Am. A 12, 2136–2144 (1995).
[CrossRef]

P. Török, P. Varga, G. Németh, “Analytical solution of the diffraction integrals and interpretation of wave-front distortion when light is focused through a planar interface between materials of mismatched refractive indices,” J. Opt. Soc. Am. A 12, 2660–2671 (1995).
[CrossRef]

S. H. Wiersma, T. D. Visser, “Defocusing of a convergingelectromagnetic wave by a plane dielectric interface,” J. Opt. Soc. Am. A 13, 320–325 (1996).
[CrossRef]

S. H. Wiersma, P. Török, T. D. Visser, P. Varga, “Comparison of different theories for focusing through a plane interface,” J. Opt. Soc. Am. A 14, 1482–1490 (1997).
[CrossRef]

D. G. Flagello, T. Milster, A. E. Rosenbluth, “Theory of high-NA imaging in homogeneous thin films,” J. Opt. Soc. Am. A 13, 53–64 (1996).
[CrossRef]

M. Mansuripur, “Distribution of light at and near the focus of high-numerical-aperture objectives: erratum; Certain computational aspects of vector diffraction problems: erratum,” J. Opt. Soc. Am. A 10, 382–383 (1993).
[CrossRef]

Jpn. J. Appl. Phys. (1)

T. Narahara, S. Kobayashi, M. Hattori, Y. Shimpuku, G. J. van den Enden, J. A. H. M. Kahlman, M. van Dijk, R. van Woudenberg, “Optical disc system for digital video recording,” Jpn. J. Appl. Phys. 39, 912–919 (2000).
[CrossRef]

Opt. Commun. (3)

J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
[CrossRef]

D. Jiang, J. J. Stamnes, “Numerical and asymptotic results for focusing of two-dimensional waves in uniaxial crystals,” Opt. Commun. 163, 55–71 (1999).
[CrossRef]

D. Jiang, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves into uniaxial crystals,” Opt. Commun. 174, 321–334 (2000).
[CrossRef]

Phys. Rev. B (1)

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

Proc. R. Soc. London Ser. A (2)

E. Wolf, “Electromagnetic diffraction in optical systems I. an integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Pure Appl. Opt. (1)

V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab: I. exact and asymptotic results,” Pure Appl. Opt. 6, 33–52 (1997).

Other (4)

A. B. Marchant, “Cover sheet aberrations in optical recording,” in Optical Disk Systems and Applications, E. V. LaBudde, ed., Proc. SPIE421, 43–49 (1983).
[CrossRef]

M. Mansuripur, The Physical Principles of Magneto-Optical Recording (Cambridge U. Press, Cambridge, UK, 1995).

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK1986).

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Figures (8)

Fig. 1
Fig. 1

Light path from the exit pupil in z=-R to the focal plane in z=0 with refraction at the interface in z=-d, the uniaxial symmetry axis aˆ, the associated wave vectors kˆ1 and kˆ2, and polarization vectors parallel to the plane of incidence pˆ1 and pˆ2 and perpendicular to the plane of incidence sˆ1 and sˆ2.

Fig. 2
Fig. 2

Intensity on a logarithmic scale (upper left), and the axial (upper right), radial (lower left), and azimuthal (lower right) flow components in the focal region for NA=0.85, n1=n2=1, Δn=0, and for circular polarization in the entrance pupil.

Fig. 3
Fig. 3

Projection of flow lines on the meridional plane (left) and on the focal plane (right) for NA=0.85, n1=n2=1, Δn=0, and =π/4 (circular polarization). Starting points of the flow lines are on the circle v=2 in the plane u=-4. The flow lines make a rotation around the optical axis of approximately π in the focal region.

Fig. 4
Fig. 4

Anomalous flow projected on the meridional plane close to the first minimum of the scalar diffraction Airy pattern. The singular point with zero Poynting vector is indicated by “S”; the point where the flow is purely azimuthal is indicated by “A.” The thin curves have zero axial flow (Sz=0) and zero azimuthal flow (Sψ=0). The line with zero radial flow (Sr=0) is dotted and coincides with the focal plane u=0. Flow lines within the toroidal region defined by the flow lines through S spiral around the circle through A.

Fig. 5
Fig. 5

Strehl ratio for NA=0.85, λ=400 nm, n1=1, n2=1.5, and d=100 μm, according to the analytical approximation [Eq. (98)] (solid curve) and according to exact numerical calculations (dashed curve).

Fig. 6
Fig. 6

Focus shift for NA=0.85, λ=400 nm, n1=1, n2=1.5, and d=100 μm, according to the analytical approximation [Eq. (97)] (solid curve) and according to exact numerical calculations (dashed curve).

Fig. 7
Fig. 7

Intensity distribution in the meridional plane for Δn=0 (upper left), Δn=5×10-3 (upper right), Δn=10×10-3 (lower left), and Δn=15×10-3 (lower right). The intensity increases from black to white. The full lines are isophotes for intensities 0.50, 0.10, 0.05, 0.02, 0.01, 0.005, and 0.001; the dashed lines indicate the geometrical light cone v=un2/(n22-NA2)1/2. The parameters used in the calculation are NA=0.85, λ=400 nm, n1=1, n2=1.5, and d=100 μm.

Fig. 8
Fig. 8

Projection of flow lines on the meridional plane for circular polarization in the entrance pupil with Δn=0 (upper left), Δn=5×10-3 (upper right), Δn=10×10-3 (lower left), and Δn=15×10-3 (lower right). The parameters used in the calculation are NA=0.85, λ=400 nm, n1=1, n2=1.5, and d=100 μm. These are the same parameters used in the calculations of the intensity distributions shown in Fig. 7.

Tables (1)

Tables Icon

Table 1 Sensitivity for Vertical Birefringence for the Three Optical Recording Standardsa

Equations (100)

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E(x, y, z)=-- d2k(2π)2 E(kx, ky, z)×exp[(i(kxx+kyy)],
H(x, y, z)=-- d2k(2π)2 H(kx, ky, z)×exp[i(kxx+kyy)].
k1z=(n12k2-kx2-ky2)1/2,
kˆ1=(sin θ1 cos ϕ1, sin θ1 sin ϕ1, cos θ1).
pˆ1=(cos θ1 cos ϕ1,cos θ1 sin ϕ1,-sin θ1),
sˆ1=(-sin ϕ1, cos ϕ1,0).
E(kx, ky, z)=(Epipˆ1+Esisˆ1)×exp[ik1z(z-z1)]+(Eprpˆ1-+Esrsˆ1-)×exp[-ik1z(z-z1)],
μ0cH(kx, ky, z)=(-n1Esipˆ1+n1Episˆ1)×exp[ik1z(z-z1)]+(-n1Esrpˆ1-+n1Eprsˆ1-)×exp[-ik1z(z-z1)],
k2zo=(no2k2-kx2-ky2)1/2,
k2ze=none(ne2k2-kx2-ky2)1/2,
kˆ2=(sin θ2 cos ϕ2, sin θ2 sin ϕ2, cos θ2),
pˆ2=(cos θ2 cos ϕ2, cos θ2 sin ϕ2,-sin θ2),
sˆ2=(-sin ϕ2, cos ϕ2, 0).
E(kx, ky, z)=Ept(pˆ2+σkˆ2)×exp[ik2ze(z-z2)]+Estsˆ2×exp[ik2zo(z-z2)],
μ0cH(kx, ky, z)=-noEstpˆ2 exp[ik2zo(z-z2)]+γnoEptsˆ2 exp[ik2ze(z-z2)],
σ=(ne2-no2)sin θ2 cos θ2no2 sin2 θ2+ne2 cos2 θ2,
γ=ne(no2 sin2 θ2+ne2 cos2 θ2),
tp=2n1 cos θ1γ2n1 cos θ2+γno cos θ1,
ts=2n1 cos θ1n1 cos θ1+no cos θ2,
rp=n1 cos θ2-γno cos θ1n1 cos θ2+γno cos θ1,
rs=n1 cos θ1-no cos θ2n1 cos θ1+no cos θ2,
E(kx, ky, z2)=tpEpi(pˆ2+σkˆ2)×exp{i[k2ze(z2+d)-k1z(z1+d)]}+tsEsisˆ2×exp{i[k2zo(z2+d)-k1z(z1+d)]},
μocH(kx, ky, z2)=-notsEsipˆ2 exp{i[k2zo(z2+d)-k1z(z1+d)]}+γnotpEpisˆ2 exp{i[k2ze(z2+d)-k1z(z1+d)]}.
Eα(r2)=--d2r1Gαβ(r2, r1)Eβ(r1),
μ0cHα(r2)=--d2r1Gαβ(r2, r1)Eβ(r1),
Gαβ(r2, r1)=-- d2k(2π)2[tp(pˆ2α+σkˆ2α)pˆ1β×exp(iΦe)+tssˆ2αsˆ1β exp(iΦo)],
Gαβ(r2, r1)=-- d2k(2π)2[-notspˆ2αsˆ1β×exp(iΦo)+γnotpsˆ2αpˆ1β exp(iΦe)].
Φe=kx(x2-x1)+ky(y2-y1)+k2ze(z2+d)-k1z(z1+d),
Φo=kx(x2-x1)+ky(y2-y1)+k2zo(z2+d)-k1z(z1+d).
ΔW=(k2ze-k2zo)(d+z2)=(k2ze)2-(k2zo)2k2ze+k2zo(d+z2)(k2ze)2-(k2zo)22k2zo(d+z2),
=k(d+z2)Δn sin2 θ2cos θ2.
Gαβ(r2, r1)=-- d2k(2π)2[tp exp(iΔW)pˆ2α pˆ1β+tssˆ2αsˆ1β]exp[iΦ(k1, k2, r1, r2)],
Gαβ(r2, r1)=-- d2k(2π)2no[-ts pˆ2αsˆ1β+tp exp(iΔW)sˆ2α pˆ1β]×exp[iΦ(k1, k2, r1, r2)],
Φ(k1, k2, r1, r2)=kx(x2-x1)+ky(y2-y1)+k2z(z2+d)-k1z(z1+d),
kxs=kn1 sin θ1 cos ϕ=kn2 sin θ2 cos ϕ,
kys=kn1 sin θ1 sin ϕ=kn2 sin θ2 sin ϕ,
k1zs=kn1 cos θ1,
k2zs=kn2 cos θ2,
x2-x1=[(R-d)tan θ1+d tan θ2]cos ϕ,
y2-y1=[(R-d)tan θ1+d tan θ2]sin ϕ.
Φ(k1s, k2s, r1, r2)=kOPL(r1, r2)=-kn1(z1+d)cos θ1+kn2(z2+d)cos θ2,
Q=z2+dkn2 cos θ2-z1+dkn1 cos θ1×z2+dkn2 cos3 θ2-z1+dkn1 cos3 θ1.
Gαβ(r2, r1)=12πiQ[tp exp(iΔW)pˆ2αpˆ1β+tssˆ2αsˆ1β]exp[ikOPL(r1, r2)],
Gαβ(r2, r1)=n22πiQ[-tspˆ2αsˆ1β+tp×exp(iΔW)sˆ2αpˆ1β]×exp[ikOPL(r1, r2)].
k OPL(r1, r2)=kOPL(r1, 0)+k2s·r2.
Bp=E0(cos ϕAx+sin ϕAy),
Bs=E0(-sin ϕAx+cos ϕAy).
I=Rn12k(Q cos θ1)1/2.
Ei(r1)=I(Bppˆ1+Bssˆ1)exp[-ikOPL(r1, 0)+iW].
E(r2)=E0Riλ 0β102π sin θ1 dθ1dϕ(cos θ1)1/2×[tp exp(iWp)(cos ϕAx+sin ϕAy)pˆ2+ts exp(iWs)(-sin ϕAx+cos ϕAy)sˆ2×exp(ik2s·r2),
μ0cH(r2)=E0Riλ 0β102π sin θ1 dθ1 dϕ(cos θ1)1/2×[-n2ts exp(iWs)(-sin ϕAx+cos ϕAy)pˆ2+n2tp exp(iWp)×(cos ϕAx+sin ϕAy)sˆ2]exp(ik2s·r2).
Ex(u, v, ψ)=-iπE0N(T0/n1n2)1/2{[F0(u, v)+F2(u, v)cos(2ψ)]Ax+F2(u, v)×sin(2ψ)Ay},
Ey(u, v, ψ)=-iπE0N(T0/n1n2)1/2{F2(u, v)×sin(2ψ)Ax+[F0(u, v)-F2(u, v)×cos(2ψ)]Ay},
Ez(u, v, ψ)=-2πE0N (T0/n1n2)1/2F1(u, v)×[cos(ψ)Ax+sin(ψ)Ay].
u=NA2n2λz2,
v=NAλ(x22+y22)1/2.
Fk(u, v)=01dρρgk(ρ)Jk(2πvρ)×exp-2πiuρ21+(1-ρ2 sin2 β2)1/2.
ρ=n1 sin θ1/NA=n2 sin θ2/NA=sin θ1/sin β1=sin θ2/sin β2,
g0(ρ)=(1-ρ2 sin2 β1)-1/4 n1+n22n1[ts exp(iWs)+tp exp(iWp)(1-ρ2 sin2 β2)1/2],
g1(ρ)=(1-ρ2 sin2 β1)-1/4 n1+n22n1tp×exp(iWp)ρ sin β2,
g2(ρ)=(1-ρ2 sin2 β1)-1/4 n1+n22n1×[ts×exp(iWs)-tp exp(iWp)(1-ρ2 sin2 β2)1/2].
T0=4n1n2(n1+n2)2,
N=n1R sin2 β1λ=R sin β1(λ/NA).
g0(ρ)=exp(iWp)+exp(iWs).
I0=120 n2n1T0π2N2,
IxI0=|F0+F2 cos(2ψ)|2|Ax|2+2 Re{[F0+F2 cos(2ψ)]F2* sin(2ψ)AxAy*}+|F2|2 sin2(2ψ)|Ay|2=12[|F0|2+|F2|2+2 Re(F0F2*)cos(2ψ)]×(|Ax|2+|Ay|2)+12[|F0|2+|F2|2 cos(4ψ)+2 Re(F0F2*)cos(2ψ)](|Ax|2-|Ay|2)+[Re(F0F2*)sin(2ψ)+12|F2|2 sin(4ψ)]×Re(Ax*Ay)-Im(F0F2*)sin(2ψ)Im(Ax*Ay),
IyI0=|F2|2 sin2(2ψ)|Ax|2+2 Re{[F0-F2×cos(2ψ)]F2* sin(2ψ)Ax*Ay}+|F0-F2 cos(2ψ)|2|Ay|2=12[|F0|2+|F2|2-2 Re(F0F2*)cos(2ψ)]×(|Ax|2+|Ay|2)-12[|F0|2+|F2|2 cos(4ψ)-2 Re(F0F2*)cos(2ψ)]×(|Ax|2-|Ay|2)+[Re(F0F2*)sin(2ψ)-12|F2|2 sin(4ψ)]Re(Ax*Ay)+Im(F0F2*)sin(2ψ)Im(Ax*Ay),
IzI0=4|F1|2|cos(ψ)Ax+sin(ψ)Ay|2=2|F1|2(|Ax|2+|Ay|2)+2|F1|2×cos(2ψ)(|Ax|2-|Ay|2)+4|F1|2 sin(2ψ)Re(Ax*Ay).
S0=|Ax|2+|Ay|2=1,
S1=|Ax|2-|Ay|2=cos(2)cos(2θ),
S2=2 Re(Ax*Ay)=cos(2)sin(2θ),
S3=2 Im(Ax*Ay)=sin(2).
IxI0=12|F0|2[1+cos(2)cos(2θ)]+Re(F0F2*)×[cos(2)cos(2ψ-2θ)+cos(2ψ)]-Im(F0F2*)sin(2)sin(2ψ)+12 |F2|2×[1+cos(2)cos(4ψ-4θ)],
IyI0=12|F0|2[1-cos(2)cos(2θ)]+Re(F0F2*)×[cos(2)cos(2ψ-2θ)-cos(2ψ)]+Im(F0F2*)sin(2)sin(2ψ)+12|F2|2×[1-cos(2)cos(4ψ-4θ)],
IzI0=2|F1|2+2|F1|2 cos(2)cos(2ψ-2θ).
II0=|F0|2+2|F1|2+|F2|2+2[|F1|2+Re(F0F2*)]cos(2)cos(2ψ-2θ).
IxI0=12(|F0|2+|F2|2)+Re(F0F2*)cos(2ψ)-Im(F0F2*)sin(2ψ),
IyI0=12(|F0|2+|F2|2)-Re(F0F2*)cos(2ψ)+Im(F0F2*)sin(2ψ),
IzI0=2|F1|2.
II0=|F0|2+2|F1|2+|F2|2,
S=Re(E×H*).
μ0cSx2I0=Re(-2i{F2 sin(2ψ)Ax+[F0-F2 cos(2ψ)]Ay}F1*[-sin(ψ)Ax*+cos(ψ)Ay*]+2iF1[cos(ψ)Ax+sin(ψ)Ay]{[F0*-F2* cos(2ψ)]Ax*-F2* sin(2ψ)Ay*})=Im[(F0-F2)F1*-F1(F0*-F2*)]cos(ψ)-Re[(F0+F2)F1*+F1(F0*+F2*)]sin(2)sin(ψ)-Im(F0F1*+F1F0*)cos(2)cos(ψ-2θ)+Im(F0F2*+F2F0*)cos(2)cos(3ψ-2θ),
μ0cSy2I0=Re(-2iF1[cos(ψ)Ax+sin(ψ)Ay]×{F2* sin(2ψ)Ax*-[F0*+F2* cos(2ψ)]Ay*}+2i{[F0+F2 cos(2ψ)]Ax+F2 sin(2ψ)Ay}F1*[-sin(ψ)Ax*+cos(ψ)Ay*])=Im[(F0-F2)F1*-F1(F0*-F2*)]sin(ψ)+Re[(F0+F2)F1*+F1(F0*+F2*)]sin(2)cos(ψ)+Im[F0F1*+F1F0*]cos(2)sin(ψ-2θ)+Im[F0F2*+F2F0*]cos(2)sin(3ψ-2θ),
μ0cSz2I0=Re({[F0+F2 cos(2ψ)]Ax+F2 sin(2ψ)Ay}×{[F0*-F2* cos(2ψ)]Ax*-F2* sin(2ψ)Ay*}-{F2 sin(2ψ)Ax+[F0-F2 cos(2ψ)]Ay}×{F2* sin(2ψ)Ax*-[F0*+F2* cos(2ψ)]Ay*})=Re(F0F0*-F2F2*)-Re(F0F2*-F2F0*)cos(2)cos(2ψ-θ).
μ0cSr2I0=cos(ψ) μ0cSx2I0+sin(ψ) μ0cSy2I0=Im[(F0-F2)F1*-F1(F0*-F2*)]-Im[(F0-F2)F1*+F1(F0*-F2*)]×cos(2)cos(2ψ-2θ),
μ0cSψ2I0=-sin(ψ) μ0cSx2I0+cos(ψ) μ0cSy2I0=Re[(F0+F2)F1*+F1(F0*+F2*)] sin(2)+Im[(F0+F2)F1*+F1(F0*+F2*)]×cos(2)sin(2ψ-2θ),
μ0cSz2I0=Re(F0F0*-F2F2*)-Re(F0F2*-F2F0*)×cos(2)cos(2ψ-2θ).
μ0cSr2I0=2 Im[(F0-F2)F1*],
μ0cSψ2I0=2 Re[(F0+F2)F1*]sin(2),
μ0cSz2I0=|F0|2-|F2|2.
drdt=l μ0cS(r)2I0,
Ws=0,
Wp=ΔW=kdΔnρ2NA2/n22(1-ρ2NA2/n22)1/2.
ΔW=kdΔn(NA/n2)2ρ2+.
F0(u,0)=01dρρ1+πi(2p-u)ρ2-12π2×(2p-u)2ρ4+1-πiuρ2-12π2u2ρ4=1+i π2(p-u)-π212[(2p-u)2+u2],
p=dΔnλ NA2n22.
S=1-π26(2p-u)2+π26u2-π24(p-u)2=1-π212(5p2-2pu+u2).
Δz=dΔnn2.
S=1-π23p2=1-112 2πdΔnλ2NAn24.
dΔndΔndl=125 λ2π n2NA2.

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